Fix an arithmetic progression $R=(a, a+m, a+2m, \ldots)$, and assume that $gcd(a,m)=1$. Define $q_R(n)$ as the following coefficients: $$\prod_{i=0}^\infty (1+ t^{a+mi}) = \sum_{n=0}^\infty q_R(n) t^n $$ In other words, $q_R(n)$ is number integer partitions of $n$ into distinct parts from $R$.

**Problem 1.** Prove that $q_R(n)$ are increasing for $\ n\ge n(a,m)$ large enough.

I first assumed this is either standard, well known, or easily follows from the existing results. Now I am less sure. My literature search gives only papers like this (A. Tripathi, "Coin exchange problem for arithmetic progressions"). Note that for $a=m=1$, we get the usual partitions into distinct parts and the claim follows from Euler's theorem that they are equinumerous with partitions into odd parts.

More generally, I need to prove that all finite differences are positive for large enough $n$. Formally, define $$(t-1)^r \prod_{i=0}^\infty (1+ t^{a+mi}) = \sum_{n=0}^\infty q_R(n,r) t^n $$

**Problem 2.** For every $r\ge 1$, prove that $q_R(n,r)>0$ for $\ n\geq n(a,m,r)$ large enough.

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